An improved convergence rate of Glimm scheme for general systems of hyperbolic conservation laws

Jiale Hua Tong Yang

Analysis of PDEs mathscidoc:1912.43992

Journal of Differential Equations, 231, (1), 92-107, 2006.12
We study the convergence rate of Glimm scheme for general systems of hyperbolic conservation laws without the assumption that each characteristic field is either genuinely nonlinear or linearly degenerate. We first give a sharper estimate of the error arising from the wave tracing argument by a careful analysis of the interaction between small waves. With this key estimate, the convergence rate is shown to be o (1) s 1 3| ln s| 1+ , which is sharper compared to o (1) s 1 4| ln s| given in [T. Yang, Convergence rate of Glimm scheme for general systems of hyperbolic conservation laws, Taiwanese J. Math. 7 (2)(2003) 195205]. However, it is still slower than o (1) s 1 2| ln s| given in [A. Bressan, A. Marson, Error bounds for a deterministic version of the Glimm scheme, Arch. Ration. Mech. Anal. 142 (2)(1998) 155176] for systems with each characteristic field being genuinely nonlinear or linearly degenerate. Here s is the
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@inproceedings{jiale2006an,
  title={An improved convergence rate of Glimm scheme for general systems of hyperbolic conservation laws},
  author={Jiale Hua, and Tong Yang},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191224211113642664556},
  booktitle={Journal of Differential Equations},
  volume={231},
  number={1},
  pages={92-107},
  year={2006},
}
Jiale Hua, and Tong Yang. An improved convergence rate of Glimm scheme for general systems of hyperbolic conservation laws. 2006. Vol. 231. In Journal of Differential Equations. pp.92-107. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191224211113642664556.
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